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%I #19 Mar 31 2021 20:23:39
%S 0,-1,2,-5,6,-13,10,-29,6,-69,-38,-197,-250,-669,-1142,-2509,-4762,
%T -9813,-19302,-38965,-77530,-155501,-310518,-621565,-1242554,-2485733,
%U -4970790,-9942309,-19883834,-39768509,-79536118
%N a(n) = 2^n * Sum_{k=0..n} k^p*q^k, where p=2, q=-1/2.
%C The factor 2^n (i.e., |1/q|^n) is present to keep the values integer.
%C See also A232600 and references therein for integer values of q.
%H Stanislav Sykora, <a href="/A232603/b232603.txt">Table of n, a(n) for n = 0..1000</a>
%H S. Sykora, <a href="http://dx.doi.org/10.3247/SL1Math06.002">Finite and Infinite Sums of the Power Series (k^p)(x^k)</a>, DOI 10.3247/SL1Math06.002, Section V.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (-1,3,5,2).
%F a(n) = ((-1)^n*(9*n^2+12*n+2) - 2^(n+1))/27.
%F G.f.: x*(-1+x)/( (1-2*x)*(1+x)^3 ). - _R. J. Mathar_, Nov 23 2014
%F E.g.f.: (1/27)*(-2*exp(2*x) + (2 -21*x +9*x^2)*exp(-x)). - _G. C. Greubel_, Mar 31 2021
%F a(n) = - a(n-1) + 3*a(n-2) + 5*a(n-3) + 2*a(n-4). - _Wesley Ivan Hurt_, Mar 31 2021
%e a(3) = 2^3 * [0^2/2^0 - 1^2/2^1 + 2^2/2^2 - 3^2/2^3] = -5.
%p A232603:= n-> ((-1)^n*(2+12*n+9*n^2) -2^(n+1))/27; seq(A232603(n), n=0..35); # _G. C. Greubel_, Mar 31 2021
%t LinearRecurrence[{-1,3,5,2}, {0,-1,2,-5}, 35] (* _G. C. Greubel_, Mar 31 2021 *)
%o (PARI) a(n)=((-1)^n*(9*n^2+12*n+2)-2^(n+1))/27;
%o (Magma) [((-1)^n*(2+12*n+9*n^2) -2^(n+1))/27: n in [0..30]]; // _G. C. Greubel_, Mar 31 2021
%o (Sage) [((-1)^n*(2+12*n+9*n^2) -2^(n+1))/27 for n in (0..30)] # _G. C. Greubel_, Mar 31 2021
%Y Cf. A001045 (p=0,q=-1/2), A053088 (p=1,q=-1/2), A232604 (p=3,q=-1/2), A000225 (p=0,q=1/2), A000295 and A125128 (p=1,q=1/2), A047520 (p=2,q=1/2), A213575 (p=3,q=1/2), A232599 (p=3,q=-1), A232600 (p=1,q=-2), A232601 (p=2,q=-2), A232602 (p=3,q=-2).
%K sign,easy
%O 0,3
%A _Stanislav Sykora_, Nov 27 2013
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